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The LOGISTIC Procedure |
and it has an asymptotic distribution with
r degrees of freedom under H0, where r is
the number of restrictions imposed on
by H0.
Suppose there are s explanatory effects of interest. The full model has a parameter vector
where
are
intercept parameters, and
are
slope parameters for the explanatory
effects. Consider the null hypothesis
where t < s.
For the reduced model with t explanatory effects, let
be
the MLEs of the unknown intercept
parameters, and let
be the
MLEs of the unknown slope parameters.
The residual chi-square is the chi-square
score statistic
testing the null hypothesis H0; that is, the residual
chi-square is
where .
The residual chi-square has an asymptotic chi-square distribution with s-t degrees of freedom. A special case is the global score chi-square, where the reduced model consists of the k intercepts and no explanatory effects. The global score statistic is displayed in the "Model-Fitting Information and Testing Global Null Hypothesis BETA=0" table. The table is not produced when the NOFIT option is used, but the global score statistic is displayed.
Suppose that k intercepts and t explanatory variables (say v1, ... , vt) have been fitted to a model and that vt+1 is another explanatory variable of interest. Consider a full model with the k intercepts and t+1 explanatory variables (v1, ... ,vt,vt+1) and a reduced model with vt+1 excluded. The significance of vt+1 adjusted for v1, ... ,vt can be determined by comparing the corresponding residual chi-square with a chi-square distribution with one degree of freedom.
For this test the number of response levels, k+1, is assumed to be strictly greater than 2. Let Y be the response variable taking values 1, ... , k, k+1. Suppose there are s explanatory variables. Consider the general cumulative model without making the parallel lines assumption
where g(.) is the link function, and
is a vector of unknown parameters consisting of an intercept
and
s slope parameters
.The parameter vector for this general cumulative model is
Under the null hypothesis of parallelism
,there is a single common slope parameter for each of the s
explanatory variables. Let
be the common
slope parameters.
Let
and
be the
MLEs of the intercept parameters
and the common slope parameters .
Then, under H0, the MLE of
is
and the chi-squared score statistic
has an asymptotic chi-square distribution with
s(k-1) degrees of freedom.
This tests the parallel lines assumption by testing the equality of
separate slope parameters simultaneously for all explanatory variables.
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